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dc.contributor.authorPfeifle, Julián
dc.contributor.authorZiegler, Günter M.
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II
dc.date.accessioned2010-06-18T18:23:45Z
dc.date.available2010-06-18T18:23:45Z
dc.date.created2004
dc.date.issued2004
dc.identifier.citationPfeifle, J.; Ziegler, G. M. On the monotone upper bound problem. "Experimental mathematics", 2004, vol. 13, núm. 1, p. 1-11.
dc.identifier.issn1058-6458
dc.identifier.urihttp://hdl.handle.net/2117/7737
dc.description.abstractThe Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n) ≤ Mubt(d,n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d,n) ≤ Mubt(d,n) holds with equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d + 2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d=6 a polytope with n=9 facets can have Mubt(6,9)=30 vertices, but not more than 27 ≤ M(6,9) ≤ 29 vertices can lie on a strictly-increasing edge-path. The proof involves classification results about neighborly polytopes, Kalai’s (1988) concept of abstract objective functions, the Holt-Klee conditions (1998), explicit enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances, as well as non-realizability proofs via a version of the Farkas lemma.
dc.format.extent11 p.
dc.language.isoeng
dc.rightsAttribution-NonCommercial-NoDerivs 3.0 Spain
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria
dc.subject.lcshPolytopes
dc.subject.lcshCombinatory logic
dc.subject.lcshGraph theory
dc.titleOn the monotone upper bound problem
dc.typeArticle
dc.subject.lemacPolitops
dc.subject.lemacCombinatoria
dc.subject.lemacGrafs, Teoria de
dc.contributor.groupUniversitat Politècnica de Catalunya. MD - Matemàtica Discreta
dc.rights.accessOpen Access
drac.iddocument2510367
dc.description.versionPostprint (published version)
upcommons.citation.authorPfeifle, J.; Ziegler, G. M.
upcommons.citation.publishedtrue
upcommons.citation.publicationNameExperimental mathematics
upcommons.citation.volume13
upcommons.citation.number1
upcommons.citation.startingPage1
upcommons.citation.endingPage11


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